Definition:Jordan Arc

Definition

Let $\tuple {x_1, y_1}, \tuple {x_2, y_2} \in \R^2$.

Let $f: \closedint 0 1 \to \R^2$ be a path from $\tuple {x_1, y_1}$ to $\tuple {x_2, y_2}$.

Then $f$ is a Jordan arc if and only if $f$ is an injection, except that we allow the possibility $\tuple {x_1, y_1} = \tuple {x_2, y_2}$.

Also defined as

In many sources, a Jordan arc $f$ is defined as a path that is an injection, so the initial point of $f$ is different from the final point of $f$.

That is, $f$ is a homeomorphism of the closed unit interval $\closedint 0 1$.

Also see

• Definition:Jordan Curve

Source of Name

This entry was named for Marie Ennemond Camille Jordan.